Andriy Lysenko, Ukraine's National Security and Defense Council spokesman, said 45,000 Russian troops are on the border, as well as tanks, aircraft, rocket launchers and artillery, adding artillery and rocket shells were fired into Ukraine Monday.
Other sources, although noting the increase in personnel, place the number lower.
"In early August, Russia significantly increased the number of troops in the vicinity of the Russian border. Our current assessment is that around 20,000 troops are now in the area. This troop presence includes tanks, infantry, artillery, air defense systems, as well as logistics troops, special forces, and various aircraft," a NATO statement said.
A five-day military drill, involving 100 planes and helicopters, is in progress within Russian territory, some of it near the border with Ukraine.
Ukrainian forces are advancing on the rebel strongholds of Donetsk and Luhansk, and officials in Kiev have expressed concern of Russian military involvement.